Abstract Algebra Answers

A relation R is defined on the set of integers as xRy iff (x+y) is even. which of the following statement is true ?
a) R is not an equivalence relation
b) R is an equivalence relation having one equivalence class
c) R is an equivalence relation having two equivalence class
d) R is an equivalence relation having three equivalence class

Let R be a k-algebra where k is a field, and M,N be left Rmodules, with dimkM <∞. It is known that, for any field extension K ⊇ k, the natural map θ : (HomR(M,N))K → HomRK(MK,NK) is an isomorphism of K-vector spaces. Replacing the hypothesis dimkM < ∞ by “M is a finitely presented R-module,” give a basis-free proof for the fact that θ is a K-isomorphism.

If K ⊇ k is a splitting field for a finite-dimensional k-algebra R, does it follow that K is also a splitting field for any quotient algebra R of R?

Let R be a finite-dimensional k-algebra and let L ⊇ K ⊇ k be fields. Assume that L is a splitting field for R. Show that K is a splitting field for R iff, for every simple left RL-module M, there exists a (simple) left RK-module U such that UL ∼ M.

Show that for any finite-dimensional k-algebra R and any field extension K ⊇ k, (rad R)K ⊆ rad(RK).

For a finite-dimensional k-algebra R, let T(R) = rad R + [R,R], where [R,R] denotes the subgroup of R generated by ab − ba for all a, b ∈ R. Assume that k has characteristic p > 0. Show that T(R) ⊆ {a ∈ R : a^p^m ∈ [R,R] for some m ≥ 1}, with equality if k is a splitting field for R.

For any group G, let Δ(G) = {g ∈ G: [G : CG(g)] < ∞}, and Δ+(G) = {g ∈ Δ(G) : g has finite order}. Show that Δ(G)/Δ+(G) is torsion-free abelian.

For any group G, let Δ(G) = {g ∈ G: [G : CG(g)] < ∞}, and Δ+(G) = {g ∈ Δ(G) : g has finite order}.
Show that Δ+(G) is a characteristic subgroup of G and that Δ+(G) is the union of all finite normal subgroups of G.

If G is an f. c. group and is finitely generated, show that [G,G] is finite.

Let G be an f. c. group. Show that [G,G] is torsion